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Theorem recexprlemex 7820

Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7821. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemex	⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

**Proof of Theorem recexprlemex**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemss1l 7818	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) ⊆ (1^st ‘1_P))
3	1	recexprlem1ssl 7816	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘1_P) ⊆ (1^st ‘(𝐴 ·_P 𝐵)))
4	2, 3	eqssd 3241	. 2 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P))
5	1	recexprlemss1u 7819	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) ⊆ (2^nd ‘1_P))
6	1	recexprlem1ssu 7817	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘1_P) ⊆ (2^nd ‘(𝐴 ·_P 𝐵)))
7	5, 6	eqssd 3241	. 2 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))
8	1	recexprlempr 7815	. . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
9		mulclpr 7755	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
10	8, 9	mpdan 421	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) ∈ P)
11		1pr 7737	. . 3 ⊢ 1_P ∈ P
12		preqlu 7655	. . 3 ⊢ (((𝐴 ·_P 𝐵) ∈ P ∧ 1_P ∈ P) → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
13	10, 11, 12	sylancl 413	. 2 ⊢ (𝐴 ∈ P → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
14	4, 7, 13	mpbir2and 950	1 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∃wex 1538 ∈ wcel 2200 {cab 2215 ⟨cop 3669 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 1^st c1st 6282 2^nd c2nd 6283 *_Qcrq 7467 <_Q cltq 7468 Pcnp 7474 1_Pc1p 7475 ·_P cmp 7477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-eprel 4379 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-1o 6560 df-2o 6561 df-oadd 6564 df-omul 6565 df-er 6678 df-ec 6680 df-qs 6684 df-ni 7487 df-pli 7488 df-mi 7489 df-lti 7490 df-plpq 7527 df-mpq 7528 df-enq 7530 df-nqqs 7531 df-plqqs 7532 df-mqqs 7533 df-1nqqs 7534 df-rq 7535 df-ltnqqs 7536 df-enq0 7607 df-nq0 7608 df-0nq0 7609 df-plq0 7610 df-mq0 7611 df-inp 7649 df-i1p 7650 df-imp 7652

This theorem is referenced by: recexpr 7821

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